Quadrilaterals and Polygons Lesson Test

Answer: D

Correct answer: 720 degrees. Use

with n = 6. Then S = 4 times 180 = 720 degrees.

• A: This is the total of the exterior angles of any convex polygon, not the interior sum of a hexagon.
• B: This multiplies 6 by 180 without first subtracting 2 from the number of sides.
• C: This subtracts 3 instead of 2 from the number of sides; 540 is the interior sum of a pentagon.
• D (correct): A hexagon has 6 sides, so the sum is (6 - 2) times 180 = 720.

Answer: A

Correct answer: 91 degrees. The interior angles of a quadrilateral total 360 degrees. The known angles add to

so the fourth angle is

• A (correct): The four angles total 360, and 360 - 269 = 91.
• B: This is the sum of the three known angles, not the missing angle.
• C: This subtracts from 540, the angle sum of a pentagon, instead of 360.
• D: This comes from a subtraction slip; 360 - 269 is 91, not 81.

Answer: D

Correct answer: 28 inches. Since

each side is 7 inches. The perimeter is

• A: This multiplies the area by 4 instead of multiplying the side length by 4.
• B: This counts only two sides of the square, 7 + 7.
• C: This is the length of one side, not the distance all the way around.
• D (correct): The side is the square root of 49, which is 7 inches, and 4 times 7 is 28.

Answer: B

Correct answer: 64 square meters. Use

The bases sum to 16, half of that is 8, and 8 times 8 = 64 square meters.

• A: This simply adds the three measurements; area requires multiplying.
• B (correct): The average base is (5 + 11) / 2 = 8, and 8 times the height 8 is 64.
• C: This multiplies all three measurements together instead of averaging the bases.
• D: This multiplies the sum of the bases by the height and forgets the one-half.

Answer: C

Correct answer: 135 degrees. An octagon has 8 sides, so its angle sum is (8 - 2) times 180 = 1080 degrees. Divided equally among 8 angles, each is

• A: This is the total of all eight interior angles, not one of them.
• B: This is the interior angle of a regular hexagon, not an octagon.
• C (correct): The interior sum is 6 times 180 = 1080, and 1080 divided by 8 is 135.
• D: This is each exterior angle, 360 / 8; the interior angle is 180 - 45.

Answer: A

Correct answer: 156 degrees. At each vertex the interior and exterior angles add to 180 degrees, so each interior angle is

As a check, the polygon has

sides, and 13 times

• A (correct): An interior angle and its exterior angle form a straight line, so 180 - 24 = 156.
• B: This is the number of sides, 360 / 24 = 15, not an angle measure.
• C: This subtracts from 360, but an interior and exterior pair adds to 180, and a convex angle is under 180.
• D: This is the sum of all 15 interior angles, (15 - 2) times 180, not a single angle.

Answer: B

Correct answer: 80 degrees. Angles W and X are consecutive, so they add to 180 degrees:

giving 5x = 150 and x = 30. Angle

and angle W = 100 degrees.

• A: This is angle W, 3(30) + 10, not angle X.
• B (correct): Neighboring angles add to 180, so 5x + 30 = 180, x = 30, and angle X = 2(30) + 20 = 80.
• C: This sets the two angles equal, but W and X are neighboring angles, which add to 180 rather than match.
• D: This is the value of x, not the angle measure.

Answer: B

Correct answer: 84 square feet. Use

with the perpendicular height: A = 14 times 6 = 84 square feet. The 9-foot slanted side is not needed.

• A: This is the perimeter, 2(14 + 9), a length rather than an area.
• B (correct): Area is base times perpendicular height, 14 times 6 = 84.
• C: This takes half of base times height, a rule for triangles, not parallelograms.
• D: This uses the slanted 9-foot side as the height; the height must be perpendicular to the base.

Answer: A

Correct answer: 40 cm. The diagonals of a rhombus bisect each other at right angles, so each side is the hypotenuse of a right triangle with legs 8 and 6. Each side is

and the perimeter is 4 times 10 = 40 cm.

• A (correct): The diagonals bisect each other at right angles, making legs of 8 and 6, so each side is 10 and the perimeter is 40.
• B: This is the area, one-half of 16 times 12, not the perimeter.
• C: This uses the full diagonals 16 and 12 as legs, giving a side of 20 instead of 10.
• D: This doubles the sum of the diagonals as if they were the sides.

Answer: C

Correct answer: 46 feet. In a figure with only horizontal and vertical edges, the top edges total the bottom edges and the left edges total the right edges. So the patio's perimeter equals that of the 14 by 9 rectangle:

• A: This is the area in square feet, 126 - 12, not the perimeter.
• B: This adds the notch's 4 and 3 on top of the full perimeter, counting those lengths twice.
• C (correct): The notch moves edges inward without changing their total, so the perimeter equals the full rectangle's, 2(14 + 9) = 46.
• D: This subtracts the notch's 4 and 3 from the perimeter, but those edges are replaced by inside edges of the same length.

Answer: D

Correct answer: 136 square feet. A border of width 2 adds 4 to each dimension, so the outer rectangle is 16 ft by 22 ft, an area of 352 square feet. The garden is 12 times 18 = 216 square feet, so the path covers

• A: This adds the 2-foot width only once to each dimension (14 by 20) instead of twice.
• B: This multiplies the garden's perimeter, 60 feet, by the path width and misses the corner squares.
• C: This is the area of the outer rectangle, which still includes the garden.
• D (correct): The outer rectangle is 16 by 22 = 352, and subtracting the 216-square-foot garden leaves 136.

Answer: B

Correct answer: $720. The area is 18 times 12 = 216 square feet. Since 1 square yard is 9 square feet, that is

The cost is 24 times $30 = $720.

• A: This prices square feet at the square-yard rate, skipping the conversion entirely.
• B (correct): The room is 216 square feet, which is 216 / 9 = 24 square yards, and 24 times $30 is $720.
• C: This divides by 27, the cubic-yard factor, which is for volume, not area.
• D: This divides by 3 instead of 9; a square yard is 3 feet by 3 feet, or 9 square feet.

Answer: D

Correct answer: 150 square cm. With side ratio

areas scale by

The larger area is

square centimeters.

• A: This squares the ratio but uses it upside down (9/25), shrinking the area.
• B: This multiplies by 3/5, using the side ratio upside down, which makes the larger figure smaller.
• C: This scales the area by 5/3, the side ratio, instead of its square.
• D (correct): Areas scale by the square of the side ratio, so 54 times 25/9 = 150.

Answer: C

Correct answer: 7 inches. Use

84 = one-half times (9 + 15) times h = 12h. So

• A: This divides the area by the shorter base alone and ignores the other base.
• B: This divides the area by the longer base alone, as if the shape were a parallelogram with base 15.
• C (correct): The average base is 12, so 84 = 12h and h = 7.
• D: This drops the one-half, solving 84 = 24h with the full sum of the bases.

Answer: B

Correct answer: 128 degrees. A pentagon's angles total (5 - 2) times 180 = 540 degrees. Adding the expressions gives

so 5x = 440 and x = 88. The largest angle is

the angles 88, 98, 108, 118, and 128 sum to 540.

• A: This is one angle of a regular pentagon, but these angles are not equal.
• B (correct): Setting 5x + 100 = 540 gives x = 88, so the largest angle is 88 + 40 = 128.
• C: This is x, the smallest angle, not the largest.
• D: This uses 360 as the angle sum, which applies to a quadrilateral, not a pentagon.

Answer: D

Correct answer: 150 degrees. From

and 12 times 9 = 108, so n = 12. The angle sum is 10 times 180 = 1800 degrees, so each angle is

• A: This is the exterior angle of the 12-sided polygon, 360 / 12, not the interior angle.
• B: This forgets to divide by 2 in the diagonal formula, solving n(n - 3) = 54 to get 9 sides.
• C: This is the sum of all 12 interior angles, not the measure of each one.
• D (correct): Solving n(n - 3)/2 = 54 gives n = 12, and a regular 12-gon has interior angles of 1800 / 12 = 150.

Answer: C

Correct answer: 144 square feet. Split the wall into two pieces and add. The rectangle is 12 times 8 = 96 square feet. The triangle has base 12 and perpendicular height 8, so its area is

The total is

• A: This leaves out the one-half in the triangle's area, counting it as 96.
• B: This counts only the rectangle and leaves out the triangular gable.
• C (correct): The rectangle is 96 and the triangle is one-half of 12 times 8 = 48, for 144 in all.
• D: This uses the 10-foot sloped edge as the triangle's height; the height is the 8-foot vertical distance.

Answer: C

Correct answer: A rectangle that is not a square. Diagonals that bisect each other mean the figure is a parallelogram. A parallelogram with equal diagonals is a rectangle. Since the diagonals are not perpendicular, it cannot be a square, so it is a rectangle that is not a square.

• A: A rhombus's diagonals cross at right angles, and a non-square rhombus has unequal diagonals.
• B: A kite's diagonals are perpendicular, and generally only one of them is cut in half.
• C (correct): Bisecting diagonals make it a parallelogram, equal diagonals make it a rectangle, and non-perpendicular diagonals rule out a square.
• D: A square's diagonals are equal and bisect each other, but they are also perpendicular, which this figure lacks.

Answer: A

Correct answer: 3 square yards. A kite has perpendicular diagonals, so its area is half the product of the diagonals.

Since 1 square yard equals 9 square feet, 27 divided by 9 is 3 square yards.

• A (correct): Half of 9 times 6 is 27 square feet, and 27 divided by 9 square feet per square yard gives 3 square yards.
• B: This drops the one-half in the diagonal formula, using 54 square feet instead of 27 before converting.
• C: This divides 27 square feet by 3, using the length factor instead of the area factor of 9.
• D: This is the area in square feet; the conversion to square yards was skipped.

Answer: A

Correct answer: About 260 square cm. Use

The perimeter is 6 times 10 = 60 cm, so

which rounds to about 260 square centimeters.

• A (correct): The perimeter is 60, and one-half of 8.66 times 60 is about 259.8.
• B: This multiplies the apothem by the perimeter and forgets the one-half.
• C: This uses the side length 10 in place of the apothem, but the apothem is the perpendicular distance from center to side.
• D: This uses one side, 10, in place of the full perimeter of 60.

Answer: A

Correct answer: 35. An interior angle and its exterior angle add to 180 degrees, so

giving e = 36 degrees. The exterior angles total 360, so

sides.

• A (correct): Find the exterior angle (36 degrees), the number of sides (10), then apply n(n - 3)/2 = 10 times 7 over 2.
• B: This is the number of sides, not the number of diagonals; the question needs one more step.
• C: This is n(n - 3) without dividing by 2, so every diagonal is counted twice.
• D: This is n(n - 1)/2, which counts every pair of vertices, including the 10 pairs joined by sides.

Answer: D

Correct answer: 1620 degrees. From one vertex, diagonals split an n-sided polygon into n - 2 triangles of 180 degrees each.

• A: This multiplies 180 by all 11 sides; the formula uses n - 2, not n.
• B: This uses n - 1 = 10 triangles instead of n - 2 = 9.
• C: This uses n - 3 = 8 triangles; n - 3 belongs to the diagonal formula, not the angle sum.
• D (correct): With n = 11, there are 9 triangles, and 9 times 180 is 1620 degrees.

Answer: A

Correct answer: 109 degrees. Neighboring angles of a parallelogram add to 180 degrees. Let the smaller angle be x:

so 2x = 142 and x = 71. The larger angle is

• A (correct): Neighboring angles are supplementary, so x + x + 38 = 180 gives x = 71, and the larger angle is 109 degrees.
• B: This is the smaller angle; the question asks for the larger one.
• C: This is 180 - 38, which is twice the smaller angle, not either angle of the parallelogram.
• D: This adds 38 to 90 as if the smaller angle were a right angle, which nothing in the problem says.

Answer: C

Correct answer: A rhombus. A parallelogram with four equal sides is a rhombus. Because its angles are not right angles, it cannot be a square, and with unequal angles it is not a rectangle either.

• A: A rectangle must have four right angles, and its neighboring sides need not be equal.
• B: A kite's equal sides come in two neighboring pairs, and a kite is generally not a parallelogram.
• C (correct): A rhombus is a parallelogram with all four sides equal; its angles need not be 90 degrees.
• D: Even books that let a trapezoid have two parallel pairs would not call it the most specific name; equal sides make this figure a rhombus.

Answer: C

Correct answer: 72 square feet. The strip runs along both ends of each dimension, so the carpet is

by

covering 108 square feet. The room is 15 times 12 = 180 square feet, so the bare floor is

• A: This subtracts the 1.5-foot strip from each dimension only once (13.5 by 10.5), forgetting the strip on the opposite wall.
• B: This is the area of the carpet itself, not the bare floor around it.
• C (correct): Subtract 2 times 1.5 = 3 feet from each dimension, then take 180 - 108 = 72 square feet.
• D: This multiplies the room's perimeter, 54 feet, by 1.5 feet, which counts the four corner squares twice.

Answer: B

Correct answer: 95 degrees. The four angles of any quadrilateral total 360 degrees. Angles B and D share what is left:

so each is

• A: This is the combined measure of angles B and D; it still has to be split between the two equal angles.
• B (correct): The two equal angles share 360 - 170 = 190 degrees, so each is 95 degrees.
• C: This assumes angle B matches angle A, but in a kite the equal pair is the two angles between unequal sides.
• D: This is 180 - 95, the supplement of the correct angle; kite angles are not supplementary in pairs like a parallelogram's.

Answer: D

Correct answer: 72 square inches. Perimeters scale by the side ratio, so

Areas scale by k squared.

• A: This multiplies the area by k = 3/2; areas scale by k squared, not k.
• B: This multiplies by k cubed (27/8), which is how volumes of similar solids scale, not areas.
• C: This adds the 12-inch difference in perimeters to the area; similar figures scale by multiplying, not adding.
• D (correct): The side ratio is 3/2, so the area is multiplied by 9/4: 32 times 9/4 = 72 square inches.

Answer: D

Correct answer: 24 square feet. Area of a rectangle is length times width: 6 times 4 = 24 square feet. Covering a surface is an area question, so the units are square feet.

• A: This is the perimeter value, 2 times (6 + 4) = 20, which measures the edge, not the surface.
• B: The number is right, but area must be given in square units, not plain feet.
• C: This adds the length and width instead of multiplying them.
• D (correct): Area is length times width, 6 times 4, in square units.

Answer: B

Correct answer: 360 degrees. Going once around any convex polygon, the exterior angles always total 360 degrees, no matter how many sides it has.

• A: This is the interior angle sum, (15 - 2) times 180; the question asks for exterior angles.
• B (correct): The exterior angles of every convex polygon add to 360 degrees, regardless of the number of sides.
• C: This multiplies 360 by 15 sides; 360 is already the total for all the exterior angles together.
• D: This is the sum of the interior and exterior angle at a single vertex, not the total around the polygon.

Answer: A

Correct answer: 360 square inches. The diagonals of a rhombus are perpendicular, so the area is half the product of the full diagonals.

The sign covers 360 square inches.

• A (correct): Half of 30 times 24 is 360 square inches.
• B: This multiplies the diagonals but forgets the factor of one half.
• C: This divides the product by 4, as if half-diagonals were multiplied and then halved again.
• D: This adds the two diagonals instead of multiplying them.

Answer: A

Correct answer: 120 square cm. Each side is

The diagonals bisect each other at right angles, forming right triangles with hypotenuse 13 and one leg 12. The other leg is the square root of

so the second diagonal is 10 cm.

• A (correct): The side is 13, half-diagonals are 12 and 5, so the diagonals are 24 and 10 and the area is half of 240.
• B: This multiplies the diagonals 24 and 10 but leaves out the factor of one half.
• C: This multiplies the side, 13, by the diagonal, 24; a diagonal is not the height of the rhombus.
• D: This multiplies the half-diagonals 12 and 5, which gives the area of only two of the four right triangles.

Answer: D

Correct answer: 112 degrees. The bases are parallel, so along each leg the two angles add to 180 degrees. Each angle at the shorter base is

and in an isosceles trapezoid the two angles at the same base are equal.

• A: This is the angle at the longer base; the angles at the other base are its supplements.
• B: This is 90 - 68, treating the angles as complementary instead of supplementary.
• C: This doubles 68; the two angles along a leg add to 180, they are not in a 2-to-1 ratio.
• D (correct): Angles along one leg between parallel bases are supplementary: 180 - 68 = 112 degrees.

Answer: C

Correct answer: 59 feet. Edging follows the boundary, so add all five sides:

• A: This leaves out one of the 9-foot sides; a pentagon has five sides to add.
• B: The sum is right, but perimeter is a length measured in feet, not square feet.
• C (correct): The perimeter is the sum of all five sides, 59 feet.
• D: This doubles the sum, a habit from the rectangle formula that does not apply when every side is listed.

Answer: A

Correct answer: 72 m. The notched side becomes

m of edge. The other three sides are 20, 12 and 12 m. The perimeter is

A notch in the middle of a side adds its two 4 m depths to the original 64 m perimeter.

• A (correct): The original 64 m perimeter gains the two 4 m inner walls of the notch: 64 + 8 = 72 m.
• B: This uses the corner-notch shortcut, but a notch in the middle of a side adds two new edges, its depths.
• C: This adds only one 4 m depth; the notch has an inner wall on each side.
• D: This removes the 6 m opening but forgets that the notch's bottom edge and two depths replace it.

Answer: B

Correct answer: 16 feet. Trim follows the boundary, so find the perimeter: 2 times

• A: This is the area value, 5 times 3; trim goes around the edge, so it needs perimeter.
• B (correct): The perimeter is 2 times (5 + 3) = 16 feet.
• C: This adds one length and one width but forgets the other two sides.
• D: This doubles the area value instead of adding the four sides.

Answer: B

Correct answer: 135 degrees. A hexagon's angles total (6 - 2) times 180 = 720 degrees. So

6x = 630 and x = 105. The largest angles are

• A: This is x, the measure of the smaller angles; the question asks for x + 30.
• B (correct): With a 720-degree sum, 6x + 90 = 720 gives x = 105, so the larger angles are 135 degrees.
• C: This uses a 900-degree sum, which belongs to a 7-sided polygon, giving x = 135 and x + 30 = 165.
• D: This is one angle of a regular hexagon, but these angles are not all equal.

Answer: D

Correct answer: 27. Each of the 9 vertices joins to

non-neighboring vertices, and each diagonal is counted from both ends.

• A: This forgets to divide by 2, counting each diagonal once from each end.
• B: This uses n(n - 1)/2, which also counts the 9 sides as if they were diagonals.
• C: This is the number of diagonals from a single vertex, not the total for the polygon.
• D (correct): Using n(n - 3)/2 with n = 9 gives 54 / 2 = 27.

Answer: C

Correct answer: 26 square yards. Floor area minus the island: 21 times 12 = 252 square feet, and

Since 1 square yard = 9 square feet,

• A: This divides by 3 instead of 9; a square yard is 3 feet by 3 feet, or 9 square feet.
• B: This converts the full 252 square feet but forgets to remove the island.
• C (correct): Subtract the 18-square-foot island from 252 square feet, then divide 234 by 9.
• D: This is the tiled area in square feet; it was never converted to square yards.

Answer: B

Correct answer: 168 square meters. Rectangle: 15 times 8 = 120 square meters. Trapezoid: half of (15 + 9) times 4 = 12 times 4 = 48 square meters. Total:

• A: This forgets the one half in the trapezoid formula, giving 96 instead of 48.
• B (correct): Add the rectangle, 120, to the trapezoid, one half times 24 times 4 = 48.
• C: This treats the trapezoid as a 15 by 4 rectangle, using only the longer base.
• D: This treats the trapezoid as a 9 by 4 rectangle, using only the shorter base.

Answer: C

Correct answer: Concave. A convex polygon has every interior angle under 180 degrees. An angle over 180 degrees makes a corner point inward, creating a dent, so the polygon is concave.

• A: A convex polygon must have every interior angle below 180 degrees, and 200 is above that.
• B: A regular polygon is convex with all angles equal, so it cannot have an angle above 180 degrees.
• C (correct): An interior angle above 180 degrees creates an inward dent, which is what makes a polygon concave.
• D: A polygon may have an angle over 180 degrees as long as its straight sides close up without crossing.

Answer: B

Correct answer: 9 sides. Each exterior angle is

The exterior angles total 360 degrees, so

sides.

• A: This subtracts 2 from the correct 9, mixing in the n - 2 from the angle-sum formula.
• B (correct): The exterior angle is 40 degrees, and 360 / 40 = 9.
• C: This stops at the exterior angle, 40 degrees, and reports it as the number of sides.
• D: This halves the exterior angle to 20 degrees before dividing 360 by it.

Answer: A

Correct answer: 120 degrees. The angles of a quadrilateral total 360 degrees. So

and x = 30. The largest angle is 4x = 120 degrees.

• A (correct): With 12x = 360, x = 30, so the largest angle, 4x, is 120 degrees.
• B: This is the value of x, not the largest angle, 4x.
• C: This is 3x, one of the two middle-sized angles, not the largest.
• D: This sets the sum equal to 180, the triangle total, giving x = 15 and 4x = 60.

Answer: C

Correct answer: 450 square feet. Start with the whole yard and remove both cutouts: 30 times 20 = 600, the pool is 120 and the shed is 30. Grass:

• A: This removes the pool but forgets to remove the shed.
• B: This removes the shed but forgets to remove the pool.
• C (correct): The yard's 600 square feet minus the 120-square-foot pool and 30-square-foot shed leaves 450.
• D: This adds the pool and shed to the yard instead of subtracting them.

Answer: D

Correct answer: 16 m. Set up the trapezoid formula and solve for the missing base.

Multiply both sides by 2 and divide by 9:

so b = 16 m.

• A: This is the sum of both bases; the known 10 m base still has to be subtracted.
• B: This forgets the one half: 117 / 9 = 13 is taken as the base sum, and 13 - 10 = 3.
• C: This divides the area by the height as if the figure were a rectangle.
• D (correct): Doubling 117 and dividing by 9 gives 26 for the sum of the bases, and 26 - 10 = 16.